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Hamiltonian systems with symmetry, coadjoint orbits and plasma physics

Marsden, Jerrold E. and Ratiu, T. and Schmid, R. and Spencer, R. G. and Weinstein, Alan J. (1983) Hamiltonian systems with symmetry, coadjoint orbits and plasma physics. Atti della Accademia delle scienze di Torino, 117 (1). pp. 289-340. ISSN 0001-4419.

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The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the Lie-Poisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the Maxwell-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. First, the Poisson-Vlasov equations are shown to be in Hamiltonian form relative to the Lie-Poisson bracket on the dual of the (nite dimensional) Lie algebra of innitesimal canonical transformations. Then we write Maxwell's equations in Hamiltonian form using the canonical symplectic structure on the phase space of the electromagnetic elds, regarded as a gauge theory. In the last step we couple these two systems via the reduction procedure for interacting systems. We also show that two other standard models in plasma physics, ideal MHD and two- uid electrodynamics, can be written in Hamiltonian form using similar group theoretic techniques.

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Ratiu, T.0000-0003-1972-5768
Additional Information:© 1983, AMS. We thank Darryl Holm, Allan Kaufman, Boris Kuperschmidt, and Shlomo Sternberg for many helpful discussions and comments. Lecture delivered by R. Schmid. Research partially supported by the Miller Institute and DOE Contract DE-ATO3-82ER12097, and the Miller Institute. Research partially supported by NSF grant MCS-81-01642.
Funding AgencyGrant Number
Miller InstituteUNSPECIFIED
Department of Energy (DOE)DE-ATO3-82ER12097
NSFMCS 81·01642
Series Name:Atti Accademia delle Science di Torino
Issue or Number:1
Record Number:CaltechAUTHORS:20100922-080914569
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20083
Deposited By: Ruth Sustaita
Deposited On:22 Sep 2010 18:24
Last Modified:09 Mar 2020 13:19

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